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Mechanism Design, Matching Theory and the Stable Roommates Problem

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Mechanism Design, Matching Theory and the Stable Roommates Problem Mechanism Design, Matching Theory and the Stable Roommates Problem Presented to the Department of Economics in partial fulfillment of the degree of Bachelor of Arts with Honors in Economics by Yashaswi Mohanty Advisor: Timothy Hubbard Second Reader: Dale Skrien Department of Economics Colby College Abstract This thesis consists of two independent albeit related chapters. The first chapter introduces concepts from mechanism design and matching theory, and discusses potential applications of this theory, particularly in relation to dorm allocations in colleges. The second chapter investigates a subset of the dorm allocation problem, namely that of matching roommates. In particular, the paper looks at the probability of solvability of random instances of the stable roommates game under the condition that preferences are not completely random and exogenous but endogenously determined through a dependence on room choice. These prob- abilities are estimated using Monte-Carlo simulations and then compared with probabilities of solving a completely random instance of the stable roommates game. Our results portray a complex relationship between the two probabilities, suggesting avenues for future research. Acknowledgements While an honors thesis is ostensibly an individual endeavour, designed to help a student develop his or her ability to conduct independent research, this project has been far from an independent affair. Although I have spent several solitary hours embedded in academic limbo, struggling to figure out a concept or getting my code to work, many of my break- throughs have been facilitated by the supporters of this project. While a number of people have contributed to the end product that you see here, a few deserve special mention. First and foremost, I must thank my advisor Prof. Tim Hubbard, who first guided me towards matching theory and its various applications. He introduced me to the possibility of designing a mechanism for college room allotment. Pondering over this problem, I stum- bled across the question that I eventually attempt to answer in this thesis. Tim has been instrumental in helping me tackle broad problems in the goal and vision of the project while simultaneously assisting with me with the intricate details of the modeling process. Perhaps his greatest contribution to this project has been suggesting that I use Kendall's tau when I was struggling to come up with a measure of rank correlation. Without this little insight, my project could have possibly collapsed! Next, I would like thank Prof. Samara Gunter, whose painstaking efforts to ensure that the entire thesis cohort was making consistent progress on their projects was an important motivating force for getting this thesis across the finish line. From making sure that my pre- sentations were not overly technical and opaque to providing critical feedback for my drafts to reassuring me of my abilities when I most doubted my capability to conduct research, Sam's guidance and support has been indispensable. I am very much indebted to her for this finished product. Third, I must thank Dale Skrien for agreeing to read this paper in the capacity of a second reader and providing helpful suggestions regarding the Python implementation of Irving's algorithm. I must also acknowledge the incredible debugging skills of Kyle McDonell who spent six hours neglecting his own honors thesis to fix my (rather poorly written) code. Kyle's modifications to the Python implementation of Irving's algorithm allowed me to finally generate the results I had been looking for. Finally, I would like to acknowledge my fellow honors students, who have shared this incredible experience with me and have experienced the peaks and nadirs of this rather tumultuous journey. This process wouldn't have been half as fun without them. i Contents 1 Matching and College Housing Allotment1 1.1 An Overview of Matching............................1 1.2 Applications of Matching Theory........................2 1.3 College Housing Allotment and Stable Roommates..............3 2 Stable Roommates with Endogenous Preferences6 2.1 Introduction....................................6 2.1.1 Solvability of stable roommates instances................8 2.1.2 Partially endogenous preferences.....................9 2.2 Methods...................................... 10 2.2.1 Setting up the model............................ 10 2.2.2 Adding exogenous room preferences................... 11 2.2.3 Endogenizing roommate preferences................... 11 2.2.4 Finding theoretical values for the solvability probability........ 13 2.2.5 Monte-Carlo methods........................... 17 2.3 Results....................................... 18 2.4 Robustness Checks................................ 21 2.4.1 Increasing the sample size........................ 21 2.4.2 Looking at larger values of n ....................... 23 2.4.3 Modifying the k-factor.......................... 25 ii 2.4.4 Trying a multiplicative model...................... 27 2.5 Conclusion..................................... 29 3 Appendices 33 A Kendall's Tau................................... 33 B Irving's Algorithm................................ 36 C Bias and the Validity of Results......................... 44 D Source Code.................................... 46 iii Chapter 1 Matching and College Housing Allotment 1.1 An Overview of Matching While traditional economic theory is concerned with markets where prices play a key role in resource allocation, there exist markets in which there are no explicit prices that determine how agents are \matched" to these resources. This is particularly the case when the resources in question are other agents. Gale and Shapley (1962) provide the classic example of the marriage market, where men and women are matched to each other given that each group has preferences over the other. In such a market, there can be no explicit prices, although agents value other agents differently based on their preferences. Although a two-sided matching problem such as marriage serves as an excellent example of a market without prices, we can also have a one-sided matching market, where agents are matched to objects. Shapley and Scarf (1974) provide a model for such a market where agents are endowed with an indivisible good such as a house and have preferences over all the houses that are assigned to agents. In such a market, agents may have incentives to exchange houses in order to maximize utility. A real life example of such a situation occurs 1 on college campuses, where students change dorm rooms at the end of the year, but dorm rooms are usually not bought or sold with money. The key features which characterize a matching market include the lack of a pricing mechanism to clear the markets, the presence of heterogeneous agents and the problem of allocating indivisible resources to these agents. Economists who study these markets are often interested in designing an algorithm or a \mechanism" that generates allocations which satisfy certain important properties. This can be particularly challenging because of numerous theoretical constraints that prevent a mechanism from having all the desirable properties (see Roth and Sotomayor (1992), and Roth (1982)). We shall discuss the nature of some of these properties in section 1.3. 1.2 Applications of Matching Theory Matching is not just a theoretical curiosity; while the game theoretical formulations of match- ing problems provide unique mathematical challenges and insights, economists are interested in the applications of the theory to market design. One of the first applications for matching was provided by Roth (1984) who looked into the problem of matching medical students to hospitals for internships. This problem can be classified as a two-sided, many-to-one matching problem. The problem is two-sided be- cause the agents can be split into distinct groups, in this case hospitals and medical students. It is also two-sided in the sense that both hospitals and doctors have preferences over each other. Since many doctors are matched to a single hospital, the problem is characterized as many-to-one matching problem. In the paper, Roth provided a history of the market and its idiosyncrasies while also presenting a game theoretic model that captures the essential elements of the market. Roth and Peranson (1999) actually presented a new mechanism for matching physicians to hospitals which was adopted by the National Resident Matching Program (NRMP). 2 Matching theory was also successfully applied in the context of school choice. In many U.S. school districts, children (and their parents) are often asked to submit the preferences over which school they would like to attend. Schools too have preferences over children, usually based on factors such as proximity, whether the child has siblings at the school, and his or her grades. Abdulkadiro˘gluand S¨onmez (2003) also formalized this problem as a two-sided, many-to-one matching problem. They they suggested that districts use a variant of the matching algorithm presented by Gale and Shapley (1962). New York and Boston are two cities which adopted these recommendations and recent empirical evidence suggests that the new mechanisms are effective in increasing the number of \good" matches. 1.3 College Housing Allotment and Stable Roommates One interesting application of matching theory is in the context of assigning college students to dorm rooms. Interestingly, matching theory is applicable at two levels in this context. Firstly, the problem of students finding
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